We restate observability in a set-valued setting which generalizes the standard output equation, then a close connection to viability kernels is established. Both global and local cases are studied by means of a set of single-valuedness results. Among these, we highlight the results which are derived by monotone set-valued mappings theory and the ones using contingent and paratingent derivatives. Moreover, this new setting gives rise to the notions of maximal observability domains and minimal unobservability domains, which we characterize in the framework of antitone mappings and their fixed points.